Are neural networks statistical models?
According to Wikipedia:


  A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.


Answer to your question:

To build any neural network model we assume the train, test and validation data are coming from a probability distribution. So, if you produce a neural network model based on statistical data then the network is a statistical model. 

Moreover, neural networks' cost function is generally a parametric model and parametric modes are statistical models.  

Please look at Goodfellow's Deep Learning book chapter Deep Feedforward Networks page 174 and 175.

From Goodfellow's book 


  Fortunately, the cost functions for neural networks are more or less the same as those for other parametric models, such as linear models. In most cases, our parametric model deﬁnes a distribution $p(y \mid x; \theta)$ and we simply use the principle of maximum likelihood. 


In conclusion, ANNs (e. g. MLP, CNN, etc.) are statistical models